I've been trying to create some tone continua in Praat using the different tonal contrasts in Mandarin. Specifically, I want to set one tone (tonal contour) as one endpoint of the continuum and another tone as the other endpoint of the same continuum. Then I want to evenly divide the distance between the two contours (endpoints) to create intermediate tone contours. Based on past studies that used tone continua as the material, this can be created by using the overlap-add (PSOLA) method in Praat, but I couldn't figure out how exactly to achieve the outcome I want. I included a figure of the type of tone continuum I'm trying to create, taken from Shen & Froud (2016). Does anyone have any ideas of how I can do that or what resources there are that can be helpful? Thanks! An example of such a tone continuum, taken from Shen & Froud (2016)

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The Praat part is pretty easy-ish, if you have those pitch functions worked out. The Mandarin part is difficult. First you need to devise a model of the two pitch patterns, meaning reducing it to an equation with some numbers assigned to (a small set of) variables. This would typically be achieved by regression analysis of tokens of the two tones, which may give you three coefficients (if you ask for 3, rather than 2 or 4). You may find that 4 is necessary. The primary question here is, "do I have a good model of Tone 2; do I have a good model of Tone 3?". A higher order question is "do I have a model of the difference between a good model and a bad model?".

One way to get a model of pitch for each time (in Praat) is to average together normalized pitch values of actual tokens of the two tones (balanced appropriately). The pitch values should be normalized for F0 and time (incidentally, this whole discussion presumes that the only difference between the tones is F0 value, and nothing about phonation type, which is manifestly false for Vietnamese and other Chinese languages). With enough data, that will eliminate bumps and jiggles.

You can then re-create those pitch contours from the two equation. By varying the coefficients (sensibly) you can create a continuum.

A completely different way to do this is to persuade yourself of a particular model of what is being controlled, which by eyeballing looks to me like onset pitch, time to floor, floor pitch, and final pitch. The main problem, as I see it, is being able to say that you achieved the outcome that you want. I would not "want" to reproduce the above graph, which has numerous artifacts that are probably from specific of tokens.

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